Quantum molecular dynamics simulations of uranium at high pressure and temperature

Constant-volume quantum molecular dynamics (QMD) simulations of uranium (U) have been carried out over a range of pressures and temperatures that span the experimentally observed solid orthorhombic $\alpha \text{-U}$, body-centered-cubic (bcc), and liquid phases, using an ab initio plane-wave pseudopotential method within the generalized gradient approximation of density-functional theory. A robust U pseudopotential has been constructed for these simulations that treats the 14 valence and outer-core electrons per atom necessary to calculate accurate structural and thermodynamic properties up to 100 GPa. Its validity has been checked by comparing low-temperature results with experimental data and all-electron full-potential linear-muffin-tin-orbital calculations of several different uranium solid structures. Calculated QMD energies and pressures for the equation of state of uranium in the solid and liquid phases are given, along with results for the Grüneisen parameter and the specific heat. We also present results for the radial distribution function, bond-angle distribution function, electronic density of states, and liquid diffusion coefficient, as well as evidence for short-range order in the liquid.

Figure 1

Energies of different hypothetical phases of uranium in mRy/atom as a function of atomic volume $\left({\text{Å}}^3\right)$ relative to the observed $\alpha \text{-U}$ phase at its equilibrium volume. Included are the relative energies of the bct, bcc, hcp, and fcc structures.

Figure 2

Energy differences for uranium in the $\alpha \text{-U}$, bct, hcp, and fcc structures relative to bcc, in mRy/atom as a function of volume per atom $\left({\text{Å}}^3\right)$. The solid lines are the energy differences computed using GGA FP-LMTO with spin-orbit coupling (Refs. 20, 48). The dashed lines are our computed energy differences using a GGA pseudopotential without spin-orbit coupling.

Figure 3

The calculated constant-volume Bain path for uranium at an atomic volume of $20.75{\text{Å}}^3$, which is close to the experimental equilibrium atomic volume at ambient conditions $20.770{\text{Å}}^3$ (Ref. 56). The solid line was computed using FP-LMTO (Ref. 20), while the dashed line was calculated using our present pseudopotential. Here $c/a=1$ and $c/a=\sqrt{2}$ correspond to the bcc and fcc structures, respectively.

Figure 4

Phase diagram of uranium up to 100 GPa determined from in situ diamond-anvil-cell x-ray/laser-heating experiments (Refs. 35, 37) together with the present calculated EOS points. The circles (atomic volume: $20.45{\text{Å}}^3$), and diamonds (atomic volume: $17.53{\text{Å}}^3$) are positioned at the chosen temperatures and calculated pressures obtained using $nvt$-ensemble quantum molecular dynamics simulations. The statistical error bars in the pressures are smaller than the size of the symbols.

Figure 5

Radial distribution function $g\left(r\right)$ (the relative probability of finding an atom at a distance $r$ from another atom) for uranium calculated using QMD at an atomic volume of $20.45{\text{Å}}^3$ and the temperatures of 330 K (solid line), 1680 K (dotted-dashed line), 2150 K (dashed line), and 2990 K (thin solid line with shaded area).

Figure 6

Bond-angle distribution function $g_3\left(\theta \right)$ (the number of bond angles an atom makes with its neighbors located within a cut-off radius of $3.9{\text{Å}}^3$) for uranium from QMD simulations at an atomic volume of $20.45{\text{Å}}^3$ and temperatures of 330 K (solid line), 1680 K (dotted-dashed line), 2150 K (dashed line), and 2990 K (thin solid line with shaded area).

Figure 7

(Color online) Snapshot of the $l\text{-U}$ structure at $T=2990\text{K}$ showing a network of tetrahedra with a common vertex, which is a characteristic of SRO in the liquid.

Figure 8

MSD for the $l\text{-U}$ states versus time as calculated from the present QMD simulations at an atomic volume of $20.45{\text{Å}}^3$ and temperatures of 2150 K (solid curve) and 2990 K (dashed curve).

Figure 9

Total electronic DOS for uranium calculated from the present QMD simulations at an atomic volume of $20.45{\text{Å}}^3$ and temperatures of 2150 K (solid line) and 2990 K (dashed line). The DOS for solid $\alpha \text{-U}$ (dotted line) and bcc (dotted-dashed) structures at $T=0$ are shown for comparison. The abrupt drop in the QMD DOS at 2 eV is an artifact of using a limited number of unoccupied states. There are three main features in the each DOS: the low-lying $6s$ states around $-42\text{eV}$, the $6p$ states around $-18\text{eV}$, and the higher-energy $7s$, $6d$, and $5f$ valence states near the Fermi level.