High pressure elasticity and thermal properties of depleted uranium

Uranium, which occupies an important central position in the early actinides, is well known for its many uses. In particular, depleted uranium and its alloys are important engineering materials due to their high density. It is well known that depleted uranium can be alloyed to effect improvements in many elastic and plastic related deformation properties, to improve corrosion resistance, and to allow flexible heat treatability.^1–3 

However, most studies of uranium have focused on the low temperature properties^4–7 or high pressure/high temperature structure.⁸ Overall, such information is useful and necessary to better understand the f-shell dynamics, but does not provide all information needed for applications. This can be particularly evident through theoretical studies,⁹ which explore the elasticity and mechanical properties from a theoretical viewpoint. In contrast, experimental studies⁸ use in situ diamond anvil cell (DAC) X-ray diffraction and laser heating techniques to explore the phase diagram up to 100 GPa and high temperature. Studies of this sort help to determine the isothermal modulus and phase boundaries, but do not extend to full elasticity or thermal properties. As a result, it is necessary that high pressure techniques move beyond just X-ray diffraction.

One technique in this area is ultrasonic interferometry (UI), which discerns information on sound speeds and elastic/thermal properties, through the application of acoustic impedance generated reflections at interfaces between materials. Well established for the study of mineral species, UI involves generation of a sound wave outside the pressurized region and components of the pressure system are chosen to propagate this sound wave to the sample. By combining such studies with X-ray measurements and radiography, it is possible to directly determine sound velocities for a wide array of sample species and indirectly determine several other parameters. Further details of the technique are given in excellent references.^10–13 

In this work, we present the first ultrasonic, high-pressure study of depleted uranium. This is accomplished through the combined use of X-ray diffraction, radiography, and ultrasonic interferometry. Through the application of these techniques, we have been able to determine high pressure elastic and X-ray equations of state, elastic moduli, and a variety of thermal properties.

Depleted uranium samples (LANL, 99.5% purity) were prepared in cylindrical, metallic form, with a diameter of 0.8 mm and thickness of 0.5 mm. These samples were prepared in standard 13 mm ultrasonic cell assemblies (Fig. 1), used with the Paris–Edinburgh Press at beamline 16-BM-B of the Advanced Photon Source. Specific details on the press setup and ultrasonic experiments are described in previous reports.^13,14 Sample assemblies were loaded in an in-house constructed containment assembly¹⁴ to prevent contamination in case of cell assembly failure.

Combined ultrasonic interferometry/X-ray diffraction/X-ray radiography techniques were used to gather data on the sample as the PE press hydraulic load increased up to 5500 psi (corresponding to ≈4.5 GPa on the sample). Energy dispersive X-ray diffraction spectra have been analyzed using MDI's JADE software¹⁶ with LeBail structural refinement on both the MgO and U sample patterns at each pressure point. An example diffraction pattern and refinement for this is shown in Figure 2. In the refinement, a pseudo-Voight peak shape was assumed and the peak full-width at half-maximum values were refined for each peak individually.

Radiographs were used to determine sample lengths. Sample length (L) was determined using contrast difference at the edge of the sample, with an automated searching procedure in Igor Pro¹⁷ being used. Estimates for the sample edges are input and the system searches for the contrast shift midpoint between the sample and the surrounding material. Once the edges are mapped, the slopes are correlated between the top and bottom of the sample and a least-squares fit is used to determine the sample length. This produces an error typically less than 1%. A pressure progression for the sample length is shown in Figure 3.

Ultrasonic interferograms were used to determine round trip transit times (t). Transit times are measured using the pulse-echo overlap technique^10–13 and have an associated error of half the smallest data time step, or 0.5 ns. Together, the sample lengths and transit times are used to determine sound velocities in the material (Fig. 4).

Sound velocities are calculated as $νx=2Ltx$⁠, with x designating either p or s for compressional or shear. X-ray results were used to determine cell parameters and volumes, along with densities (ρ). Elastic moduli are determined from the compressional (longitudinal) and shear sound velocities and density as $m=ρνm2$⁠, where m is used to represent L (longitudinal modulus) or G (shear modulus). Adiabatic bulk moduli are determined at each pressure point from the other two moduli as $B=L−4/3G$⁠.

Beyond just the elastic parameters, there are methods for determining several thermal parameters from the sound velocity and X-ray results. These include the elastic Debye temperature,^18,19 Grüneisen parameter,²⁰ acoustic thermal conductivity,²¹ and the isothermal bulk modulus.¹² In particular, the elastic Debye temperature, lattice thermal conductivity, and Grüneisen parameter are computed starting from the average/mean sound velocity, which is computed as

From this mean sound velocity, the elastic Debye Temperature is determined from

with a being the atomic weight, h being the Planck constant, and ρ being the density. The Grüneisen parameter is determined directly from the sound velocities through

The elastic Debye temperature is then used to compute the Acoustic Thermal Conductivity as

with V being the unit cell volume, M the formula mass per unit cell, T being the temperature, and γ being the Grüneisen Parameter. Finally, the isothermal bulk modulus is determined as

In addition to the parameters reported in this paper, it is also possible to compute Young's modulus and the Poisson ratio from the acquired data, using standard formulae relating the Bulk and Shear moduli (or the Longitudinal moduli, if so desired).

X-ray results have been found to agree well with the reported ambient Cmcm structure.^1,22 Ambient cell parameters are listed in Table I, with the Birch–Murnaghan 3rd order equation of state parameters, resulting from the equation of state fit. These fits were performed using the EOSFit GUI package.²³ From these, the X-ray determined density is found to be 18.99(1) g/cm³. The equation of state and pressure–volume points are illustrated in Figure 5.

Least squares fits to the ultrasonic results are listed in Table II, with plots of the sound velocities with pressure being previously reported.¹⁴ These fits give an ambient shear modulus of 74 GPa and a longitudinal modulus of 231 GPa. In each case, the parameter values and pressure derivatives are illustrated in Table II and indicate varying degrees of increase with applied load (Fig. 6).

Calculation of the thermal parameters has also been performed from Eqs. (1) to (5). The results shown of this calculation are shown in Figure 7. As with the elastic results, ambient values are in Table II. Using volumetric thermal expansion,²⁴ the isothermal bulk modulus is computed to be 131.1(1) GPa and increases slowly with the applied pressure.

Previous measurements of bulk modulus for depleted uranium include static DAC measurements and shock studies.^8,25–35 However, these results are widely scattered, as is shown in Table III. In particular, it is seen from these results that the X-ray and ultrasound results from our work are consistent. It is not unusual to see variations in reported equation of state (EOS) parameters for uranium, or any other material in general, and this can be a result of a number of factors. For diamond anvil cell work, such as Akella et al.²⁸ or Yoo et al.,⁸ this variation is possibly associated with pressure medium effects. For example, pressure media are known to freeze with pressure increase. As a medium freezes, it begins to support shear waves and thus no longer provides a truly hydrostatic environment. Ultimately, as has been shown, non-hydrostatic stresses may lead to stiffer P-V behavior compared with the hydrostatic case. Such an explanation does not work in general for several reasons. As all pressure media will solidify at some pressure, there is no truly hydrostatic pressure medium. Additionally, such an argument is not applicable to the ultrasound work we present, as the pressure media are already solid. One principle effect of this is that it cannot be assumed to be purely hydrostatic. However, it is very likely to be quasi-hydrostatic and to remain so over the pressure range.

Therefore, it is more likely that such discrepancies are associated with the samples used. As such, it is difficult to comment on particular effects that could be responsible for the large range variations. One possibility is that the lower moduli are associated not with the regular α phase, but with the α₁ phase. Evidence can be found for this in some of the reported results, such as the work of Dewaele et al.,³⁴ where their reported modulus mates well with their theoretical results for the α₁ phase. Grain size could, and probably does, play a factor, as has been previously argued by Lander et al.³⁶ Other possibility includes a correlation between B₀ and $B0′$ (Ref. 37) or the purity of the sample, which is known to have effects on the transition pressure and strength of zirconium.³⁸ Further work is planned to investigate these possibilities.

The elastic Debye temperature shows a steady increase with applied pressure, indicative of stiffening of the lattice and upward shifts of the phonon frequencies. The value found from these results (327.7 K) is slightly higher than the theoretical determinations, which range from 228 to 300 K.^{4,29,33,39,40} However, this difference likely only indicates slightly stiffer bonds than those present in the calculations. Further comparison of results is presented in Table IV. In this table, we present both low-temperature limits^4,36,39 and ambient temperature values.^29,32,33,40 From these results, it is easily seen that the low-temperature results show a discrepancy between the elastically determined Debye temperature and that from heat capacity of at least 40 K. In contrast, the ambient temperature values deviate from our value (determined directly from sound velocities) by at minimum 27 K. Again, as previously discussed, it is possible and likely that the difference at the low-temperature limit remains at ambient temperature.

Similar behavior is observed for the acoustic thermal conductivity, which amounts to nearly 0.5 $Wm K$ under ambient conditions. Previously, the total thermal conductivity was reported⁴¹ to range from 20 to 30 $Wm K$⁠. It is well known that the thermal conductivity in most materials can be considered dominated by acoustic phonon and electronic conduction.⁴² However, other contributions can present the dominant portion of the thermal conduction in a material. For uranium, the electrical resistivity can be used to obtain an estimate of the electronic contribution to the thermal conductivity through the Wiedemann–Franz equation. Assuming a theoretical Lorenz number (i.e., L₀ = 2.44 μW mΩ/K²), the reported resistivity of approximately 25 μΩ cm (Ref. 43) translates into a thermal conductivity of 29.28 $Wm K$⁠, which is very close to the reported values.

Changes in bonding with pressure are also evident in the Grüneisen parameter. In this case, the decreasing trend with applied load suggests a decreasing anharmonic component. It is known that increasing temperature results in an increase of the anharmonicity exhibited by uranium metal.^44,45 However, pressure is generally expected to have the opposite effect (i.e., lowering temperature is similar to increasing pressure), in agreement with our observations. Our search of the available literature on the topic has not yielded any previous reports of this parameter for the α structure at ambient temperature. Such a result directly implies that pressure is more strongly localizing the atomic oscillations to the crystallographic site, but further exploration of this phenomenon is required to give better detail.

We have performed the first high pressure ultrasonic study of depleted uranium metal with pressure. This study has covered the pressure range up to 4.6 GPa and has resulted in combined X-ray radiographic and diffractive and ultrasonic interferometric measurements of the sample. From this, an X-ray pressure–volume curve and sound velocities have been determined.

These sound velocities and X-ray densities have been used to obtain elastic moduli. Further, various thermal parameters and conversion between the adiabatic and isothermal bulk modulus have resulted from these elastic moduli and external data.²⁴ It has been found that pressure generally has the impact of increasing material strength and elastic moduli. The sole exception to this is the decrease in the Grüneisen parameter with applied load. Overall, the parameters determined are found to be in good agreement with previous reports, where these are available.

In addition to the first set of elastic/thermal results on uranium metal at high pressure, this work has also demonstrated the capacity to make such measurements on radioactive and other potentially hazardous materials. This first application of a previously reported containment setup allows the study of radioactive materials through these techniques.¹⁴ While groundbreaking in the study of radioactive metals and compounds, future study is required to further determine the pressure–temperature evolution of such materials.

Los Alamos National Laboratory (LANL) is operated by LANS, LLC, for the DOE-NNSA under Contract No. DE-AC52-06NA25396. The authors acknowledge funding support from LANL Science Campaigns 1 and 2. Portions of this work were performed at HPCAT (Sector 16), Advanced Photon Source (APS), Argonne National Laboratory. HPCAT operations are supported by DOE-NNSA under Award No. DE-NA0001974 and DOE-BES under Award No. DE-FG02-99ER45775, with partial instrumentation funding by NSF. APS is supported by DOE-BES, under Contract No. DE-AC02-06CH11357. Use of the Advanced Photon Source, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science by Argonne National Laboratory, was supported by the U.S. DOE under Contract No. DE-AC02-06CH11357. M.K.J. gratefully acknowledges the support of the U.S. Department of Energy through the LANL/LDRD Program and the G.T. Seaborg Institute for this work. The authors also thank Dr. Baosheng Li for the use of his pulse-echo overlap software for determination of transit times.