Structures of solid hydrogen at 300 K

We present simulated x-ray diffraction patterns (XRD) from molecular dynamics studies of phase transformations in hydrogen at room temperature. Phase changes can be easily identified in simulation, by directly imaging the atoms and measuring correlation functions. We show that the room-temperature XRD patterns for hydrogen phases I, III, IV, and V are very similar. The signatures of the transformations in XRD are weak peaks and superlattice reflections denoting symmetry breaking from the hexagonal-closed-packed (hcp) phase I, and a pronounced change in the $c/a$ ratio. The XRD patterns implied by molecular dynamics calculations are very different from those arising from the static minimum enthalpy structures found by structure searching. Simulations also show that within phase I, the molecules become increasingly confined to the basal plane and suggest the possibility of an unusual critical point terminating the phase I-III boundary line. With these results, we propose a paradigm shift, i.e., that the predictions from density functional theory calculations should be seen as the most likely hypothesis. Specifically, we show that recent experimental results support the picture advanced by molecular dynamics simulations, and are inconsistent with the interpretation of an isostructural hcp transformation.

Figure 1

Single $G$ layer: all structures comprising stacking of these layers are labeled “phase III.” (Left) Schematic: red dots show position of the threefold rotation axis. Blue arrows represent molecular axes. The orientational transformation breaks the symmetry and induces a dipole, as indicated by the direction the arrow. (Right) Time-averaged positions of atoms aver 1 ps from one layer in the 180-GPa simulation using BLYP. All molecules are in plane, apparently short bonds occur when the molecule has rotated through ${180}^{\circ }$ at some stage. Notice how molecular centers are slightly displaced from hcp sites.

Figure 2

Schematic of typical in-plane atomic positions in the so-called “graphitic” ($G$) layers of phase IV. The red diamond shows $a$ and $b$ vectors for single unit cell. Each unit cell has two equivalent and one nonequivalent hexagon: $x, y$, and $z$ are used to label the layer stacking of the nonequivalent site. In this notation (a) $G_z^{\prime }$, (b) $G_x^{\prime \prime }$. The molecular “$B$” layers are like phase I and are not shown, they simply comprise a molecule at the center of each hexagon, again six atoms per layer. In static relaxation these $B$ molecules have well-defined orientation (e.g., the $Pbcn$ structure), but at room temperature they are disordered and reorient on a 100-fs timescale. Due to constraints from periodic boundary conditions, in MD simulation a two-layer cell in phase IV PT conditions adopts $B{G}_z^{\prime }$ stacking, 4 layers $B{G}_z^{\prime }B{G}_z^{\prime \prime }$, 6 layers $B{G}_x^{\prime }B{G}_y^{\prime }B{G}_z^{\prime }$, 8 layers $B{G}_z^{\prime }B{G}_z^{\prime \prime }B{G}_z^{\prime }B{G}_z^{\prime \prime }$, 12 layers $B{G}_x^{\prime }B{G}_y^{\prime }B{G}_z^{\prime }B{G}_x^{\prime }B{G}_y^{\prime }B{G}_z^{\prime }$.

Figure 3

Equation of state (volume per atom at 300 K) for all structures with BLYP (circles) and PBE (triangles) showing that functional effects are much larger than structural differences, and the uncertainty due to functional is about 20 GPa. Small blue dots are XRD data [15].

Figure 4

Plot of $c/a$ ratio for simulations in various cells. Arrows show sharp change in $c/a$ with transition to phase IV, and more gradual change approaching phase III. Note the significant functional dependence in the calculated transition pressure. The very high-pressure metallic $Cmca$ structures are twinned, so the change in “$c/a$ ratio” for the simulation cell signifies the transition, but is not the $c/a$ ratio of $Cmca$. $d$ spacings can be calculated from the $c/a$ ratio and the volume per molecule ($\sqrt{3}c{a}^2/4$): the observed $d$ spacings are $\left(100\right)=\sqrt{3}a/2, \left(101\right)=\sqrt{(3a^2/4+c^2)}$, and $\left(002\right)=c/2$.

Figure 5

Variation of orientation order parameter $\langle cos\theta \rangle$ with pressure at 300 K. Incompatibility with boundary conditions means that the transition to phase III or IV is suppressed in the red “phase II” simulations which started in $P{6}_3/m$ symmetry. Transitions were also identified from visualization of trajectories.

Figure 6

Simulated x-ray diffraction patterns for hydrogen phase II at 140 GPa. $P{6}_3/m$ and $Pca{2}_1$ are the zero-temperature ground states proposed by structure search. Black line is MD starting from $P{6}_3/m$ at 300 K temperature: MD starting from $Pca{2}_1$ or $P{6}_3/mmc$ are indistinguishable: all are hcp phase I with $hkl$ indexing as shown.

Figure 7

Simulated x-ray diffraction patterns for hydrogen phase III at 190 GPa calculated using BLYP. $B2/n$ and $P{6}_{1}22$ are the zero-temperature ground states proposed by two structure-search calculations [16, 20]. MD lines are labeled by temperature and are averaged over 2 ps starting from $P{6}_{1}22$ at various temperatures. The pattern for hcp, the proposed solution for the Ji et al. [15] room-temperature XRD patterns, is shown for comparison. In addition to the hcp-like peaks shown, the MD also predicts a superlattice reflection at a larger $d$ spacing around 2.65 Å, with $hkl=(\sqrt{\frac{1}{3}},0,\frac{1}{3})$. Ji et al. do not index such small-angle data.

Figure 8

Simulated x-ray diffraction patterns for hydrogen phase IV at DFT pressure of 220 GPa. Tick marks show XRD data from Ji et al. at equivalent density assigned to ${\left(100\right)}_{\text{hcp}}, {\left(002\right)}_{\text{hcp}}$, and ${\left(101\right)}_{\text{hcp}}$, alongside the pattern from the ${P6}_3/mmc$ structure calculated in Ji et al., and simple hcp. Our MD for the two different hexagonal candidates (Fig. 2), is averaged over 2 ps at 300 K starting from $P{6}_{1}22$ ($B{G}_{x}B{G}_{y}B{G}_z$) and starting from $Pbcn$ ($B{G}^{\prime }B{G}^{\prime \prime }$). $Cmca$(4), $Cmca$(12), and $Ibam$ are the zero-temperature ground states found by structure search [16].

Figure 9

Comparison of hexagonal $c/a$ ratio for experimental XRD [15] (magenta) with DFT: isostructural hcp $P{6}_3/mmc$ (blue squares), structure-search enthalpy minimum $P{6}_{1}22$ (black triangles), and MD (green circles). All DFT calculations used BLYP functional, and previously reported results [15, 20] were replicated in this study.