eFF, a method to simulate large scale excited electron dynamics

Thermodynamics of warm dense hydrogen

The behavior of hydrogen at extreme pressures and moderate temperatures has relevance to the phase partitioning of giant planetary interiors [34], as well as to the design of systems for inertial confinement fusion [35]. It is challenging to compute hydrogen's equation of state at high densities, as the pressure is sensitive to the thermal motions of the electrons. For example, at higher temperatures, electrons become more diffuse, which in dense matter results in higher pressures via stronger Pauli repulsion. Also, as its nuclei are pressed closer together, hydrogen may attain a metallic character [36].

We have used eFF to compute the equation of state of bulk hydrogen [1]. Over a temperature range of 0 to 100,000 K and densities up to 1 $ \mathrm{g/cm^{3}}$, we find excellent agreement with experimental, path integral Monte Carlo, and linear mixing equations of state, as well as single-shock Hugoniot curves from shock compression experiments.

At room temperature, diamond anvil apparatuses have been used to measure the pressure versus density of solid hydrogen up to 120 GPa. We computed an equation of state under the same conditions using eFF, and found good agreement with experiment [37] (Figure 19). In the figure, we indicate density with the Wigner-Seitz radius $ r_{s}$, where each atom occupies an average volume of $ \frac{4}{3} \pi r_{s}^{3}$.

Figure 19: Equation of state for solid deuterium at 300 K and varying density. Pressures from eFF dynamics are in good agreement with diamond anvil experiments.

It has been proposed that at high density and moderate temperatures ( $ \mathrm{r_{s}}$ = 2 bohr and T = 15,300 K), hydrogen might undergo a plasma phase transition, where molecular dissociation and atomic ionization occurs simultaneously [38].

Figure 20 shows the pressure versus temperature of $ \mathrm {D_{2}}$ at $ r_{s}$ = 2 bohr. We find that the eFF equation of state is in the range with the best quantum mechanics above 10,000 K, where path integral Monte Carlo is accurate [39,40]; and it agrees well with the Saumon-Chabrier equation of state below 10,000 K, which used parameters fitted to describe the molecular phase, and is expected to be accurate in that regime. The good agreement of eFF with PIMC persists up to $ \sim$100,000 K.

Figure 20: Equation of state for liquid deuterium at $ r_{s}$ = 2 bohr with varying temperature. eFF is applicable and consistent with the best theory over the entire temperature range shown (PIMC [39,40] above 10,000 K and Saumon-Chabrier equation of state [38] below 10,000 K).

We observe a gradual dissociation of molecules into separated atoms over the temperature range 10,000 K to 20,000 K, and a gradual ionization of atoms into protons and electrons at temperatures exceeding 50,000 K. While eFF describes the dissociation of molecules into atoms as the temperature increases, it does not show the discontinuous first derivative in the equation of state that would characterize a first-order phase transition.

We verified further the applicability of eFF to this hydrogen dissociation and ionization regime by reproducing single shock Hugoniot curves, and finding good agreement with the results of gas gun [41], Z machine [42], and convergence geometry experiments [43], where liquid deuterium is compressed and heated from 20 K to tens of thousands of degrees within tens of nanoseconds (Figure 21).

Figure 21: Single shock Hugoniot curve for liquid $ \mathrm {D_{2}}$. We show here that eFF agrees well with most experiments: gas gun (red dots), Z machine (green dots), and convergence geometry (orange). The validity of the Nova laser data (blue dots, [44]) has been questioned.

Hence the simple eFF, with no adjustable parameters, provides an unified and continuous description of solid molecular, liquid molecular, atomic, and ionized regimes, from 0 to 100,000 K.

Auger process in a diamond nanoparticle

We are interested in the general question of how highly excited states (hundreds of electron volts of excess energy) created by energetic photons, electrons, or reactive plasmas induce bond fragmentation which ultimately leads to surface etching. To study this, we examine as a prototype process Auger relaxation in a diamond nanoparticle (Figure 22).

Figure 22: Diamond nanoparticle $ \mathrm {C_{197} H_{112}}$ showing sites highlighted in red where carbon core electrons are individually ionized and Auger dynamics studied.

In the Auger process, an atom from which a core electron has been ionized relaxes by dropping a valence electron into the core shell; the energy difference transfers to another valence electron, which is ejected. Such processes are thought to participate in the ejection of surface species in photon and electron-stimulated desorption [45] (PSD and ESD).

We ionize core electrons at the surface of the nanoparticle and at different depths below the surface. In this way, we can study how the distance over which an Auger excitation relaxes and propagates affects the distribution of ejected electron energies and the composition of desorbed atomic species.

Figure 23 shows a single Auger trajectory where at time zero a 1s electron is removed from the central carbon. We examine the motions and energies of the valence electrons surrounding the newly-created core hole. There are four electrons with the same spin as the ionized electron, and no longer Pauli-excluded from the core, they fall in toward the nucleus together.

The electrons collapse toward the nucleus at nearly the same rate, but small differences in their initial position due to the finite temperature of the nanoparticle (300 K) ensure that one electron eventually ``wins'' and occupies the core hole, becoming an energetic core electron. The other three valence electrons bounce away from the now-filled core. Two of these electrons stay bonded to the nucleus, though highly excited, while one electron breaks free, only to be trapped in the particle $ \sim3\ \mathrm{\AA}$ away, with its energy dissipated among the other electrons of the solid (not shown in the figure).

Figure 23: Positions and energies of electrons following ionization of a carbon core electron at the center of the diamond nanoparticle. Valence electrons surrounding the core hole with the same spin as the ionized core electron are highlighted with color. (a) Distance of valence electrons from the core hole. (b) Potential energy of valence and core electrons.

In this example, the electron was trapped within the nanoparticle. But electrons can also escape. When this occurs, the electron's size increases linearly with time, as one would expect from a free particle wave packet.

Aggregating statistics over thousands of independent trajectories, we obtain the energy distribution of ejected electrons, and the composition of ejected atomic fragments, both as a function of excitation depth. As expected, electrons are released primarily from the outer layers of the nanoparticle, explaining the surface selectivity of Auger spectroscopy.

Carbon fragments can only be ejected from the outermost layer, which is not surprising. What is interesting is the hydrogen species that emerge, and their charges: protons and neutral atoms are emitted only through surface excitations, while hydrides are emitted from surface and deeper excitations.

In order to explain this observation, we examine the detailed motion of electrons in breaking carbon-hydrogen bonds to understand exactly how these bonds fragment. We uncover a story which has some consistency with existing models of ESD/PSD, such as the Knotek-Feibelman model for proton emission [46,47]; and with recent experiments that analyzed the proton, hydride, and secondary electron emission from PSD on diamond surfaces [48].

However, there are some surprises as well, which has caused us to propose an extension of existing models. We have also discovered through the simulations a new minor pathway for the desorption of protons.