Over the past decade, much progress has been made in exploring experimentally the high-pressure phases of solid elemental hydrogen and towards producing solid metallic hydrogen by extreme compression [1–11]. Dias and Silvera reported to have produced metallic hydrogen at 495 GPa (5 Mbar) and low temperatures of 5 and 83 K [12], a study that proved unusually contentious [13]. Subsequently, Loubeyre, Occelli and Dumas reported an abrupt transition to a metallic state in hydrogen at a pressure of 427 GPa, which they attributed to a molecular metallic phase rather than monatomic metallic hydrogen [14]. These reports of metallic hydrogen, based on different experimental techniques, await to be confirmed and complemented with further measurements to ascertain the nature of these states.

One of the driving forces in the quest for the creation of monatomic metallic hydrogen via extreme compression at low temperatures has been the expectation of an unusual state of matter. On the one hand, metallic hydrogen can be considered the simplest of all metals in the sense that the single electron per atom forms an electron gas permeating a proton lattice, taking the series of the simple alkali metals to the extreme limit. On the other hand, the most likely crystal structure to be adopted by hydrogen at ∼500 GPa, near the predicted transition to its monatomic phase, is the Cs-IV-type crystal structure [15–21] (space group ), as illustrated in figure 1(a). This is in marked contrast to the more densely-packed body- and face-centred cubic structures of the alkali metals at low pressures. More importantly, Ashcroft suggested long ago that monatomic metallic hydrogen is a high-temperature superconductor [22], and later estimates and calculations predicted transition temperatures ranging from ∼200 K to above room temperature [23–27]. It has furthermore been argued that a high zero-point energy and quantum behaviour of the protons at high density will hinder crystallisation and produce a low- or zero-temperature quantum fluid rather than crystalline metallic hydrogen [28]. Subsequent calculations, which took proton quantum effects into account [29–31], predicted melting temperatures of the order of 200 K at a pressure of 500 GPa.

Once dense, low-temperature metallic hydrogen becomes available for further study, an important task will be to establish whether it is a solid, and if so, determine its crystal structure. This is to provide a firm basis for understanding the metallic state and to test the structure predictions of numerous computational studies [28]. In current quasi-static high-pressure experiments, hydrogen is compressed in a diamond-anvil high-pressure cell, yielding a micron-sized sample. This is presently far too small for neutron diffraction, which is generally the preferred technique for structural studies on hydrogen-bearing solids, despite much progress in high-pressure neutron diffraction instrumentation [32]. Synchrotron x-ray diffraction using a high-brilliance, tightly focused x-ray beam is the technique of choice for crystal-structure studies in static multi-megabar experiments [33–35] and has been used to study crystal structures of molecular hydrogen phases to ∼250 GPa [11].

In all crystalline metals except hydrogen and helium, the ion cores alone produce an x-ray diffraction pattern suitable for structure determination, irrespective of the spatial distribution of the itinerant valence electrons. In monatomic metallic hydrogen, however, the entirety of the electrons forms the electron gas. As was pointed out previously [36], it is therefore not clear what the x-ray diffraction pattern of monatomic metallic hydrogen will look like. In the limiting case of the commonly (and successfully) employed free-electron Fermi gas model, the resulting hom*ogeneous electron gas would not produce a structured x-ray diffraction pattern and thereby prevent any structure determination with this method. Borinaga *et al* [26, 37] discussed the electronic band structure of metallic hydrogen (Cs-IV-type) in some detail and noted that it is not far from the free-electron limit. This poses the question: *Is it fundamentally possible to determine the crystal structure of monatomic metallic hydrogen with x-ray diffraction?*

We used first-principles electronic structure calculations in the framework of density functional theory (DFT) to study the electron density distribution in several phases of monatomic and molecular hydrogen, in particular the metallic Cs-IV-type phase at 500 GPa. We demonstrate that the electron-proton interaction leads to a significant redistribution of the electron charge, away from the hom*ogeneous-electron-gas limit and giving rise to a distinct x-ray diffraction pattern. We will compare this with results for more free-electron-like metallic hydrogen with the hypothetical fcc (face-centred cubic) structure, consider the effect of zero-point and thermal atomic motion on the predicted diffraction patterns and evaluate whether experimental patterns can be expected to contain sufficient information to discriminate between different candidate crystal structures. Finally, we present an atomic scattering factor for the hydrogen atom that is optimised for the quantitative modelling and analysis of x-ray diffraction data from monatomic metallic hydrogen at high density.

All electron-density calculations were performed in the framework of DFT and the full-potential augmented-plane-wave + local orbital (APW+lo) approach as implemented in the Wien2k code [38]. For the results reported here, exchange and correlation effects were treated with the local density approximation (LDA) [39]. Calculations using the revised generalised gradient approximation for solids (PBEsol) [40] reproduced the LDA results without any significant difference. For Cs-IV-type hydrogen at 500 GPa, the charge density calculations were found to be well converged with a 20^{3} grid for k-point sampling (635 k-points in the irreducible part of the Brillouin zone); a plane-wave basis set determined by , where bohr is the atomic sphere radius and *K*_{max} the largest plane-wave wave vector; and bohr^{−1} for the magnitude of the largest vector in the charge density Fourier expansion. The calculations for further hydrogen phases were performed with equivalent parameters. X-ray structure factors were calculated from the electron density by Fourier transformation as implemented in Wien2k.

In order to obtain the mean squared displacement, , of the atoms due to zero-point and thermal motion, lattice dynamical calculations for Cs-IV-type hydrogen and deuterium were conducted in the harmonic approximation as implemented in phonopy [41]. Interatomic force constants were determined with the finite-displacement method, using Vasp [42] for the calculation of DFT forces on the atoms in a supercell of the primitive Cs-IV-type unit cell. The DFT calculations were performed with a all-electron potential, a plane-wave energy cut-off of 1000 eV and a grid for k-point sampling. For the calculation of the mean squared displacement, phonon modes were sampled on a dense grid of wave vectors. The mean squared displacement is equivalent to the isotropic atomic displacement parameter used in crystallography, .

### 3.1.Cs-IV-type hydrogen

In the tetragonal Cs-IV structure type (space group ), four atoms occupy the Wyckoff site 4*a* with positions , , , in reduced coordinates. Starting from the structural parameters obtained in earlier computational work [16], we optimised the unit cell volume and axial ratio for a target pressure of 500 GPa. The lattice parameters thus obtained are *a* = 1.201 Å and *c* = 3.153 Å (). From the volume dependence of the total energy, a DFT static pressure of 500 GPa was obtained with both the LDA [39] and the PBEsol [40] exchange-correlation potential. The pressure contribution from the volume-dependence of the zero-point vibrational energy as calculated in the harmonic approximation (discussed below) was found to be well below 1 GPa and thus negligible.

Figures 1(b) and (c) show the calculated electron density distribution in monatomic, Cs-IV-type metallic hydrogen at 500 GPa in two planes through the unit cell. The electron density ranges from 0.49 to 3.4 *e* Å^{−3}, clearly far removed from a hom*ogeneous distribution. In fact, this electron distribution is more modulated than what a simple superposition of the electron densities of free hydrogen atoms yields, 0.63–2.5 *e* Å^{−3}. The salient features of the charge distribution in figures 1(b) and (c) can be reproduced in calculations at the Hartree level, i.e. for an independent-electron gas without exchange and correlation effects.

Figure 2 shows that the deviation from the free-electron limit is also apparent in the electronic density of states, namely in the form of a redistribution of states near the Fermi energy. The redistribution can be traced back to band splittings near the N point in the Brillouin zone, as discussed previously by Borinaga *et al* [37]. The departure from the density of states for a free-electron gas of the same density is, however, far less pronounced than in some high-pressure phases of alkali metals, e.g. lithium with the cI16 structure at ∼50 GPa [43]. The DFT and free-electron-gas values for the Fermi energy in Cs-IV-type metallic hydrogen differ by only 4%. Overall, the deviation from the free-electron limit is moderate in terms of the electron energy distribution, but very pronounced in terms of the spatial electron density distribution. This behaviour is arguably not unique to hydrogen, but it is masked in other nearly-free-electron systems, such as the archetypical sodium, by the presence of the core electrons.

Since x-ray diffraction intensities are given by the Fourier coefficients of the electron density distribution, the inhom*ogeneous electron distribution in hydrogen will give rise to a distinct diffraction pattern. This is quantified in figure 3, which shows powder diffraction intensities as obtained from the Fourier transform of the DFT electron density. Here, a short x-ray diffraction wavelength of 0.2 Å was assumed, which could be used at a synchrotron radiation source so as to avoid restricting the accessible reflections by the limited opening angle of a high-pressure cell. Even though the diffraction intensity diminishes quickly with increasing diffraction angle, there are four distinct reflections near 10° and 15°. Together, these would suffice to identify reliably the Cs-IV-type crystal structure, which has only two free parameters, the lattice parameters *a* and *c*.

Figure 3 also shows diffraction intensities calculated with the independent atom model as employed routinely in crystal structure refinements. This model can provide a reasonable estimate of the diffraction intensities if the atomic scattering factor for *bonded hydrogen* after Stewart, Davidson and Simpson (SDS) is used [44, 45]. This atomic scattering factor effectively describes an electron distribution that is more localised than in a free H atom, and it yields a much better description of the diffraction intensities than the atomic form factor for the neutral H atom [45]. We will return to this point below.

### 3.2.fcc hydrogen

With two electrons per primitive unit cell, hydrogen can be considered a semimetal, and some deviations from the density of states in the free-electron gas approximation are thus expected. We have therefore also considered the simpler case of close-packed hydrogen with the fcc structure. Even though this phase was predicted to become stable only at a much higher pressure of the order of 3.5 TPa [16], it can be expected to have an electronic structure closer to the free-electron limit than the lower-symmetry Cs-IV-type phase.

The DFT calculations for hypothetical fcc H at 500 GPa were performed using a cubic lattice parameter of *a* = 1.646 Å. The more efficient packing of the fcc structure avoids the very short contacts in the *ac* and *bc* plane of the Cs-IV-type structure (figures 1(a)–(c)). Figure 4(a) illustrates that the electron density in fcc H is distributed somewhat more evenly in the interstitial region, but still with considerable accumulation around the nuclei. As expected, the electronic density of states (up to the Fermi energy, ) is closer to the free-electron limit, see figure 4(b). However, the electron distribution is still sufficiently inhom*ogeneous so as to produce a distinct x-ray diffraction pattern as shown in figure 4(c).

### 3.3.Zero-point motion and thermal effects

Zero-point and thermal motion of the atoms reduces diffraction intensities with increasing diffraction angle. In order to quantify this effect, which may be expected to be particularly large for light hydrogen atoms [11], we calculated the isotropic atomic displacement parameter, , in hydrogen at the same density as before. Figure 5(a) shows that zero-point motion alone produces a relatively large Å^{2} in our calculations, which increases by 15% up to 400 K.

Atomic vibrations reduce the diffraction intensities discussed above by the Debye–Waller factor, . Figure 5(b) shows that zero-point and thermal motion in hydrogen reduce the intensities of the reflections near 10° and 15° by factors of about 0.6 and 0.4, respectively.

To lessen this effect, one might consider cooling the sample, but figure 5(b) shows that it has little impact because the zero-point motion is the dominant contribution. One could also consider using the heavier isotope deuterium, which does reduce the mean squared displacement by , but this translates only to a moderate gain in intensity, see figure 5(b).

Comparison with the Debye–Waller factor for silicon at ambient conditions shows that the issue here is not a particularly small Debye–Waller factor for hydrogen, as sometimes assumed [11], but rather that the atom density in high-pressure solid hydrogen is extremely high, so that already the first reflections appear at unusually large . Altogether, the effect of zero-point motion on the diffraction patterns of metallic hydrogen is not prohibitive, but it will render experiments more challenging.

Our zero-temperature results for the atomic displacement parameter, , are consistent with the zero-point mean squared displacement, , reported previously by Borinaga *et al* [26] on the basis harmonic phonon calculations similar to those in the present work. These values are approximately a factor of 2 larger than the value reported by Azadi *et al* [18] based on quantum Monte Carlo simulations, and the origin of this discrepancy is unclear. Anharmonic corrections were previously found to be small, of the order of a few per cent [26]. It was noted before that a mean vibrational amplitude of ∼20% of the nearest-neighbour distance Å, as confirmed here, is significantly larger than the threshold for melting in a classical system according to Lindemann's rule, , but below that for a quantum system with a higher threshold [46, 47] of . This is in line with classical molecular dynamics simulations [29–31] that predicted melting temperatures of ∼310–380 K at 500 GPa, whereas path integral simulations [29–31], which take quantum effects for the protons into account to some extent, arrived at lower melting temperatures of 160–250 K. Altogether, at room temperature and 500 GPa, metallic hydrogen can be expected to be close to the melting line or in the liquid phase. While this means that low temperatures may be required to obtain *solid* metallic hydrogen, it facilitates producing single crystals of metallic hydrogen through annealing near room temperature or by cooling from the melt, similar to what was achieved in sodium [48, 49] at around 1 Mbar.

### 3.4.Differentiability between structures

From the above discussion it is clear that only a few reflections may be observable in x-ray diffraction studies on metallic hydrogen, in particular if these are limited to powder-diffraction experiments. This raises the question whether such limited diffraction data would be sufficient to differentiate between the likely candidate crystal structures identified in computational studies.

In addition to the Cs-IV-type phase ( with ), we considered therefore a further monatomic and two molecular phases. McMahon and Ceperley identified a second hydrogen structure with space group but much smaller axial ratio of to be close in enthalpy to the Cs-IV-type phase, near 500 GPa [16]. This second structure is a diamond-type structure with fourfold coordination and moderate tetragonal distortion (but far from the *β*-tin structure type of the same symmetry but much stronger distortion, , and sixfold coordination [50]). Starting from the previously reported structural parameters [16], we optimised the unit cell volume and axial ratio for a target pressure of 500 GPa and obtained the lattice parameters *a* = 1.714 Å and *c* = 1.563 Å ().

The two molecular phases, labelled –4 and –12 in previous work [18, 19, 51], have four and twelve atoms in their primitive unit cells, respectively. The –12 phase was identified as a possible intermediate phase between a semiconducting molecular phase (with space group and the monatomic metallic Cs-IV-type phase [51]. –12 was predicted to be stable at around 440 GPa [51], and Loubeyre *et al* [14] identified this as the likely structure of the metallic hydrogen phase that they reported for pressures above 425 GPa. The –4 phase is a competing molecular phase at pressures around 450 GPa [18, 19, 51].

For the calculation of the electron density distribution and diffraction intensities of the –4 phase at 500 GPa, we first optimised the structure for a target pressure of 500 GPa. Orthorhombic lattice parameters of Å and fractional atomic coordinates for the Wyckoff 2*a* site of (0, 0.1174, 0.1234) were obtained. For the –12 phase, we scaled the 300 GPa lattice parameters reported in [19] to yield the same density as that of hydrogen at 500 GPa, and then optimised the lattice parameters and atomic positions for a target pressure of 500 GPa. This gave orthorhombic lattice parameters of Å and fractional atomic coordinates for the three symmetry-independent atomic sites of (0, 0.0137, 0.1505), (0, 0.1319, 0.4566) and (0, 0.2841, 0.3163).

Figure 6 shows a comparison of the calculated powder diffraction intensities of the four phases, which were obtained, as above, from the Fourier transform of the charge distribution calculated with Wien2k. The dominant feature in all patterns are the reflections with diffraction angles near 10°. The two phases have a single reflection near 10°, whereas the molecular –12 and –4 phases show a triplet and a doublet, respectively. Such splittings can be resolved with a modern synchrotron x-ray diffraction setup, unless size- or strain-broadening from the sample itself become too severe. However, even in poorly resolved powder diffraction patterns, the molecular phases should yield asymmetric peaks near 10°, which would permit them to be differentiated from the monatomic phases, and higher-angle reflections would yield additional information for identifying the crystal structure among the likely candidates from computational work.

### 3.5.Atomic scattering factor for metallic hydrogen

As seen in figures 3 and 4(c), the independent atom model can yield a reasonable estimate of the diffraction intensities if the SDS atomic scattering factor for bonded hydrogen [44, 45] is used. However, for a quantitative analysis of x-ray diffraction intensities, an atomic scattering factor optimised for hydrogen in monatomic phases is required. From the DFT-based structure factors, *F _{hkl} *, and the corresponding reduced diffraction angles, , one can obtain a set of discrete values for the atomic scattering factor

*f*as

with the geometrical structure factor

where the sum is over all atoms in the unit cell with fractional coordinates .

Figure 7(a) shows atomic scattering factors thus obtained from the two phases at 500 GPa, fcc hydrogen at the same pressure, and also fcc hydrogen at a higher density corresponding to a pressure of 1000 GPa (lattice parameter *a* = 1.499 Å). There is little variation arising from the different structures and densities, so that it is appropriate to describe these data with a common atomic scattering factor, which can be represented in terms of the usual expansion [45],

with the wavelength in units of Å. The following parameters for high-pressure monatomic hydrogen (H-HPM) were obtained by least-squares fitting:

The corresponding curve is shown by the solid line in figure 7(a). Figure 7(b) compares the atomic scattering factor for high-pressure monatomic hydrogen, H-HPM, with the conventional atomic scattering factors [44, 45] for the free H atom, H-RHF (relativistic Hartree–Fock), and for bonded hydrogen, H-SDS. It is clear that H-HPM and H-SDS both describe a more localised electron distribution than that of the neutral H atom as represented by H-RHF. This is in line with the preceding analysis of the electron density distribution in metallic hydrogen. Even though the difference between the H-SDS and H-HPM form factors may appear small in figure 7(b), it translates to significant differences in the relative diffraction intensities calculated with either form factor. The optimised atomic scattering factor, H-HPM, reduces the deviation between DFT and IAM diffraction intensities by more than two thirds compared to H-SDS. Quantitatively, the standard deviation of the DFT:IAM intensity ratios for the two phases (up to 30°, 28 reflections) decreases from 21% to 6%. The remaining deviations result from the fact that some details of the electron distribution cannot be reproduced by superposing spherically symmetric atomic charge densities. For the same reason, the independent atom model is not suitable for modelling diffraction intensities from molecular hydrogen phases adequately because a significant fraction of the electron charge is located in the covalent bonds between nuclei.

### 3.6.Experimental considerations

We have shown that it is fundamentally possible to probe the crystal structure of metallic hydrogen with x-ray diffraction and to differentiate between different candidate structures, but such an experiment will undoubtedly be challenging with the currently used approaches. Not only is it necessary to compress hydrogen in a diamond-anvil cell with a suitably large opening angle to pressures at the extreme limit of what is presently feasible for such highly compressible materials. A micrometre-sized sample of hydrogen, even at very high density, also scatters only weakly, and this weak signal will be on top a broad and intense background, mostly from Compton scattering in the diamond anvils.

X-ray diffraction measurements on polycrystalline hydrogen up to 183 GPa were reported by Akahama *et al* [4]. In such experiments, the diffraction intensity is spread out on Debye–Scherrer rings, which makes them difficult to detect on top of the Compton background. As mentioned above, the melting temperature of hydrogen was predicted to be of the order of 200 K at 500 GPa [29–31]. If accurate, it would open way to producing single-crystal samples of hydrogen at high pressure from the melt or by 'annealing' below room temperature. With that, the diffraction rings would collapse into diffraction spots and become detectable at a much better signal:background ratio, similar to what was demonstrated in the single-crystal x-ray diffraction study of hydrogen to 250 GPa by Ji *et al* [11]. In addition, it would greatly facilitate resolving reflections that are close in interplanar spacing, such as the 10° reflections discussed above (figure 6).

The background signal from the diamond anvils could be reduced through the use of a multichannel collimator system [52]. Partially perforated diamond anvils have also been used to reduce the background [53, 54], but it is doubtful whether such anvils with a blind hole on the table side would be mechanically sufficiently stable to reach the extreme pressure required here.

Using density functional calculations, we have demonstrated that monatomic metallic hydrogen has a strongly modulated electron density distribution, far from the free-electron gas limit, so that its crystal structure can in principle be identified with x-ray diffraction experiments. At the same time, the electronic energy spectrum and optical properties are not far from the free-electron limit as discussed previously [26, 37]. Zero-point motion of the hydrogen atoms produces a significant reduction in diffraction intensity, which is not prohibitive but will render experiments more challenging. However, the large zero-point motion in hydrogen may also prove beneficial in that it was predicted to lower the melting temperature of metallic hydrogen near 500 GPa to below room temperature, which opens way to producing quasi-single-crystal samples at high pressure. X-ray diffraction experiments on such samples, rather than polycrystalline samples, would help to detect the weak signal from hydrogen on top of the strong inelastic-scattering background from diamond anvils. We presented an atomic scattering factor optimised for the quantitative analysis of x-ray diffraction data from monatomic hydrogen at high pressure. Altogether, x-ray diffraction experiments can play a key role in determining under which conditions metallic hydrogen is fluid and solid, identifying its crystal structure and thereby providing a crucial ingredient for understanding its physical properties.

All data that support the findings of this study are included within the article.