Sub-100 ps laser-driven dynamic compression of solid deuterium with a ∼40 μJ laser pulse

Materials at high density are central to fundamental physics,^1–6 and planetary science.⁷ Hydrogen at high density is of great interest for fundamental properties^8,9 and inertial confinement fusion.⁵ Generally, dynamic methods employing compression waves are required to obtain the highest compressions, and facility-sized instruments have been constructed to perform large scale laser-driven dynamic compression experiments.^5,6,10–12

“Slow” compression^4,12,13 is essential to obtain high density hydrogen,⁴ since shockwave (fast) compression cannot achieve high density^6,14 due to the production of dissipative heat.^6,13 Unlike shockwave compression (which is irreversible), in the limit of slow compression the compressed material remains near equilibrium. Dynamic experiments which approach this limit have been termed any of “quasi-isentropic,”¹⁵ “ramp,”¹² or “shockless”¹³ compression.

It was recently shown conceptually that the energy required to dynamically compress a sample varies as the third power of the compression time,¹⁶ i.e., a reduction in the compression time by a factor of 10 reduces the required compression energy by a factor of 1000. In conventional laser driven dynamic compression, the compression time typically ranges from 10 to 100 ns.^10,12 Thus, for materials which equilibrate on a picosecond time scale, the drive energy for quasi-isentropic compression experiments can be orders of magnitude smaller than currently used. For example, recently Fe (which does not equilibrate rapidly) was ramp compressed to ∼1.67× initial density over <30 ns using 285 kJ of pulse energy.¹² Here, we quasi-isentropically compress solid deuterium (at an initial pressure of 24 GPa) to ∼1.24× initial density over <100 ps using <40 μJ of pulse energy. The laser drive energy per unit density change in our experiments is ∼10⁹ times smaller than the longer time scale experiments. Even accounting for the relatively small difference between the bulk modulus of Fe (170 GPa) and deuterium (70 GPa at 24 GPa pressure¹⁷), this result suggests that, for a given final density, because it equilibrates rapidly hydrogen might be compressed using orders of magnitude lower drive energy than used in longer time scale experiments. This is a substantial experimental advantage, enabling relatively high throughput,^12,18 and more extreme conditions for a given laser drive energy.

Here, we describe dynamic compression of solid deuterium in a diamond anvil cell (DAC), but these results apply broadly to dynamically compressed hydrogen outside the DAC. Usually, hydrogen has been dynamically compressed from a cryogenic liquid or solid at low pressure.¹⁹ Although we compress solid deuterium in a DAC, it is very likely that cryogenic liquid deuterium equilibrates on a faster time scale than the solid deuterium in our experiments. Boehly et al.,¹⁹ in shock compression experiments of cryogenic liquid deuterium which reach a final pressure of 2500 GPa, measure a single shock rise time (which is comparable to the equilibration time) of less than 22 ps. Thus, compression of cryogenic liquid deuterium to extreme density is likely possible with a compression time of much less than 1 ns. At least one solid, Al, is known to exhibit stiff behavior under fast compression, in contrast to our results for solid deuterium.¹³ 

Here, we apply ultrafast laser compression and diagnostic methods^{10,18,20–26} to demonstrate quasi-isentropic compression of solid deuterium in less than 100 ps, at least an order of magnitude faster than conventional quasi-isentropic dynamic compression.^6,11–13,19

We identify quasi-isentropic compression via a measurement of the compression wave speed via an oscillatory interferometric optical reflectivity signal,^18,20,27 analogous to the signal derived from picosecond acoustics.²⁷ Effectively, the compressed region acts as a scanning optical etalon, where one period of the oscillations corresponds to a change of λ/2 in the optical pathlength between the compression front and the piston, giving velocity information as described in more detail in Ref. 20. In particular, the frequency of this oscillation increases with wave speed. To obtain wave speed information for compression over a sub-100 ps time scale, the period of the observed oscillations must be sufficiently short (consistent with a sufficiently large wave speed) to observe a period of oscillation in the signal over the compression time. In part to satisfy this requirement, we obtain sufficiently fast wave speeds by precompressing the deuterium (D₂) sample in a DAC. After the initial static precompression, we dynamically compress the sample from initial pressures ranging from 22 GPa to 55 GPa at room temperature to final stresses as high as 71 GPa, over time scales of tens of picoseconds, using ultrafast laser driven dynamic compression as described in Refs. 20 and 18.

A schematic drawing of the experiment is shown in Fig. 1. A ∼1 μm aluminum ablation layer partially coats one tip of one diamond of the DAC. The initial pressure in the DAC was determined using ruby fluorescence.²⁸ A laser pump pulse was used to drive a compression wave into the Al ablation layer, which, after transiting the ablation layer, dynamically compressed the deuterium sample. The pump pulse energy measured before entering the DAC ranged between 30 and 40 μJ and the pump spot size at the ablator was ∼20 μm FWHM in diameter. The pump pulse temporal profile was a clipped gaussian, with an initial fast (∼5 ps) rise and ∼100 ps duration, similar to previous work.^{18,20,21,25,29} The peak intensity of the pump was approximately 10¹¹ W/cm². The radius of curvature of the of the compression front is much larger (>200×) than the distance over which the waves propagate, so the flow is approximately 1D.¹⁶ These experiments result in localized damage at the pump spot—each dataset was obtained in single shot, but more than one shot could be obtained from a single DAC load.

The probe comprised a pair of low energy (∼5 μJ) pulses with the same temporal characteristics as the pump, delayed with respect to each other by 5 ps, with a spot diameter on the sample of greater than 200 μm. The probe pulses are Doppler shifted upon reflection from the moving compression wave and D₂/Al (piston) interface^18,20,21 (which is directly analogous to picosecond acoustic spectroscopy²⁷). The observable is an optical phase shift between the two probe pulses resulting from these Doppler shifted reflections, as a function of time. Two examples of the raw data are shown in Fig. 2. Single shot time resolution in these experiments is obtained by spectrally resolving chirped probe pulses, where the wavelength of the probe sweeps over its bandwidth as a function of time (i.e., early times in the pulse are red, late times are blue).^18,20,21,25 Both pump and probe have an intensity bandwidth of ∼25 nm full width at half maximum. Data are obtained over a spatial 1D cut through the pump profile, and the time traces are obtained at the center of the pump profile.^18,20,25 Full details of the experimental setup are given in Refs. 18 and 20.

In general, the data are the sum of two contributions: a monotonic non-oscillatory part (the offset) which is approximated in Fig. 2 by green traces, and a quasi-sinusoidal oscillation centered vertically at the offset. When the Gladstone-Dale approximation³⁰ is valid (as it is here),³¹ the particle speed at the piston is given by the offset via $particle speed=offset×λ/(4 π)$⁠, as shown in the right side axes of the plots. Given the measured particle speed, the shockwave speed can be derived from the oscillation in the data.²⁰ 

There are two types of dynamic response that can, in principle, occur upon fast, longitudinal, high strain compression of a soft solid.^32,33 Rapid compression (i.e., faster than the relaxation rate) of the sample generates a large longitudinal elastic compression wave near the piston interface, followed after some relaxation time by plastic deformation.^29,34 Slower compression generates a quasi-isentropic compression wave which travels at the wave speed given by the bulk modulus (analogous to the wave speed in a material without shear strength, such as a liquid).³⁵ Critically, longitudinal elastic waves speeds in deuterium are more than 25% faster than bulk waves¹⁷ at the pressures of our experiment. So, longitudinal elastic waves are readily distinguishable from bulk compression waves via characterization of the wave speed. For compression that is faster than relaxation, unrelaxed elastic stress in the sample near the piston is comparable to the peak stress applied to the material,²⁹ i.e., large. Thus, for a solid, a compression wave with a bulk wave speed near the piston interface (where these experiments are performed) indicates that the equilibration rate is larger than the compression rate, i.e., quasi-isentropic compression.

To characterize the wave speed distribution of compression waves in our experiment, we fit the data to the simulated signal given by a conventional quasi-isentropic compression model of the compression wave.³⁵ This model assumes the sample behaves as a fluid without shear strength, i.e., the wave speed is given by the bulk modulus of the sample, rather than a longitudinal elastic modulus (e.g., Young's modulus). An example of this model (corresponding to the fit in Fig. 2(a)) is shown in Fig. 3. The thermodynamic isentrope, in combination with a particle speed boundary condition at the piston interface completely specifies the fluid flow.³⁵ In particular, the wave speed of a given pressure-density state (i.e., the slope of an isobar in Fig. 3) is the sum of the local sound and particle speeds, where the local sound speed is given by $(∂P/∂ρ)S$⁠, and the particle speed along an isobar is set by the piston speed boundary condition. We use “quasi-isentropic” to refer to compression by a wave whose speed is derived from the bulk sound speed of a material along the thermodynamic isentrope, consistent with the conventional definition of the speed of sound in a material without strength given above. The pressure-density for a given particle speed is derived from well-known expressions in Ref. 35. To derive the optical response, we use the Gladstone-Dale relationship³⁰ to obtain the index of refraction from the density given by the fluid flow model, and use only first order reflections from the compression wave front to determine the simulated signal.²⁰ 

We calculated the isentrope from 24 GPa initial pressure using statistical mechanics variational methods³⁶ calibrated by measured data, including sound speeds.¹⁷ This calculated isentrope gives bulk sound speeds to within 1% of aggregate bulk speeds in experimental sound speed data (which extend to 24 GPa).¹⁷ To model the signal for fits, we assume the slope of the calculated isentrope (in wave speed vs. particle speed space, where the isentrope is well approximated by a linear function) and use the bulk sound speed at the initial pressure as a fitting parameter. The particle speed boundary condition comprises the remaining fitting parameters, and is approximated by a piecewise linear function (e.g., green traces in Fig. 2(a)), a form which has been used to analyze previous experimental data from Al.^10,37–39

The particle speed boundary condition used in the fit must be well-centered vertically on the oscillations in the signal (see, for example, Fig. 2(a)), which strongly constrains the fit. As shown in Fig. 2(a) and other data taken from ∼25 GPa initial pressure,⁴⁰ the fits match the data well.

For quasi-isentropic compression, the wave speed of the ramp varies from the bulk sound speed at the initial pressure (and zero initial particle speed) up to the final pressure and peak wave speed (at the final particle speed and pressure-density). Fig. 4 shows the peak wave speed vs. peak pressure given by fits to the deuterium data of Fig. 2(a) and other data.⁴⁰ These data are consistent with wave speeds corresponding to quasi-isentropic compression, rather than longitudinal elastic compression and indicate that, from an initial pressure of 25 GPa, we observe quasi-isentropic dynamic compression of solid deuterium to a final pressure as high as 46 GPa in less than 100 ps.

In contrast, as shown in Fig. 4, at initial pressures of more than 50 GPa (see Fig. 2(b) for an example of the raw data), we observe wave speeds consistent with longitudinal elastic compression. Data from samples at initial pressures above 50 GPa are qualitatively similar to shock data^18,20 and different from quasi-isentropic data (as shown, for example, in Fig. 2(a)). When fit using the shock wave model of Ref. 20, these data indicate elastic shock wave speeds which are consistent with longitudinal elastic wave speeds¹⁷ and inconsistent with the calculated quasi-isentropic compression curve from 53 GPa initial pressure (see Fig. 4), indicating elastic shock compression in this precompression regime. At the piston, similar to previous observations of deformation in thin samples,^29,37,38 the elastic wave is very large, and may be expected to be faster than the (low amplitude) longitudinal acoustic wave speed at the initial pressure, consistent with our observation. For comparison to longitudinal acoustic wave speeds, it is reasonable to assume our measured longitudinal wave speed corresponds to an effective stress bounded by the initial and final stress in the sample, as shown in Fig. 4. Acoustic wave speeds have not previously been experimentally measured in deuterium at static compressions greater than 24 GPa. These results also indicate that a sustained longitudinal elastic stress of at least 16 GPa (see Fig. 4) can be maintained for at least 100 ps (see Fig. 2(b)) in deuterium starting from a static pressure of 55 GPa. We note that changes near the end of the trace in Fig. 2(b) may indicate the start of relaxation of the elastically compressed state, but, due to edge artifacts exhibited in this type of data, there was no useful data beyond the end of what is plotted.

36 GPa compression data⁴⁰ cannot be fit with either a quasi-isentropic model or an elastic shock model, possibly due to kinetics resulting from the relaxation of anisotropic stress on a time scale comparable to the compression time scale (which neither model takes into account).

We speculate that the abrupt deviation in quasi-isentropic data from fits (as shown in Fig. 2(a) and other data⁴⁰), typically around 50–80 ps after the wave arrival in the sample, is due to the loss of coherence in the wave front upon the convergence of isobars—the ramp to shock transition.⁴¹ Such a transition was observed in all near 25 GPa initial pressure data, but not observed in data corresponding to initial pressures above 50 GPa. The convergence of isobars in models (for data at near 25 GPa initial pressure) corresponds very well to the timing of the transition in the corresponding deuterium time domain data.

In conclusion, we observe compression wave speeds consistent with quasi-isentropic compression of deuterium over 100 ps to 1.24× initial density from 25 GPa initial pressure. At initial pressures higher than 50 GPa, we observe elastic shocks, indicating that material relaxation has not occurred on the experimental time scale. These results strongly suggest that cryogenic liquid hydrogen can be quasi-isentropically compressed on an ultrafast time scale, thus providing a route to high density hydrogen using orders of magnitude less laser drive energy than required for longer time scale dynamic compression.

We acknowledge useful conversations with L. E. Fried, E. J. Reed, B. W. Reed, W. J. Nellis, J. Eggert, G. Collins, R. Smith, M. Howard, B. Militzer, J. Carter, R. Hemley, H. Radousky, and J. Forbes. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344 with Laboratory directed Research and Development funding (11ERD039), as well as being based on work supported as part of the EFree, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Grant No. DESC0001057.