First-principles calculation of H vibrational excitations at a dislocation core of Pd

Palladium is an ideal system for understanding the behavior of hydrogen in metals. In Pd, H is located both in octahedral sites and in dislocation cores, which act as nanoscale H traps and form Cottrell atmospheres. Adjacent to a dislocation core, H experiences the largest possible distortion in $\alpha \text{-Pd}$. Ab initio density-functional theory computes the potential energy for a hydrogen in an octahedral site in $\alpha \text{-Pd}$ and in a trap site at the core of a partial of an edge dislocation. The Pd partial dislocation core changes the environment for H, distorting the H-Pd bonding which changes the local potential, vibrational spectra, and inelastic form factor for an isolated H atom. The decrease in excitation energy is consistent with experiments, and the calculations predict distortions to the H wave function.

Figure 1

(Color online) (top) Strain at octahedral sites in Pd edge-dislocation core. The edge dislocation has a total Burgers vector $\frac{a}{2}\left[110\right]$ while the partials have Burgers vector $\frac{a}{6}\left[211\right]$ and $\frac{a}{6}\left[12\overline{1}\right]$ and are separated by $6.5b$. (bottom) Hydrogen (small red) in the octahedral site below the Pd partial dislocation core in the $\left(1\overline{1}\overline{1}\right)$ slip-plane. The hydrogen is in an octahedral site with large tensile strain; the top three Pd atoms are the most expanded, and the bottom three are less expanded. The dislocation core displaces the neighboring Pd atoms for the octahedral site from the relaxed bulk positions shown by the $⟨100⟩$ arrows. Moreover, the hydrogen atom moves away from the top three atoms ([100], $\left[00\overline{1}\right]$, and $\left[0\overline{1}0\right]$) increasing the distance to $1.96–2.16\text{Å}$. The hydrogen moves closer to the other three atoms, decreasing the distance to $1.87–1.91\text{Å}$, compared to a distance of $1.96\text{Å}$ for H in an octahedral site in fcc-Pd.

Figure 2

(Color online) Hydrogen potential energy as a function of displacement along (a) $⟨100⟩$, (b) $⟨111⟩$, (c) $⟨110⟩$ directions in the slip plane, and (d) $⟨110⟩$ directions out of the slip plane. Thick black shows response in $\alpha \text{-Pd}$, and thin black shows response in 5% expanded $\alpha \text{-Pd}$. Positive (negative) displacements along corresponding directions are dashed (solid). Arrows show corresponding displacements, for the potential curves of the same color and linestyle. Thick displacement arrows have positive out-of-page components; negative for thin arrows. Single points on the plots are the results of individual first-principles calculations, and the continuous curves are fit to the Hellmann-Feynman forces. (a) The cube directions [100] (circles), [010] (squares), and [001] (diamonds) show the greatest change to the potential due to the dislocation core. The softest directions, [100] and $\left[0\overline{1}0\right]$, point toward regions of expansion while the $\left[\overline{1}00\right]$ and [010] stiff directions bring the hydrogen closer to Pd atoms. (b) The octahedral directions [111] (circles), $\left[\overline{1}11\right]$ (squares), $\left[1\overline{1}1\right]$ (diamonds), and $\left[11\overline{1}\right]$ (triangles) show less change due to the dislocation core. The most significant softening is for the octahedral direction corresponding to the slip-plane normal. (c) The out-of-plane closed-packed directions $\left[10\overline{1}\right]$ (squares), [011] (diamonds), and $\left[\overline{1}10\right]$ (triangles) are similarly softened in the direction of expansion and stiffened in the opposite direction compared with bulk. (d) The in-plane closed-packed directions [101] (squares), [110] (diamonds), and $\left[0\overline{1}1\right]$ (triangles) are all stiffened compared with bulk.

Figure 3

(Color online) Inelastic form factor $S\left(\vec{q}\right)$ for a H adjacent to the dislocation core and in octahedral bulk sites. The momentum $\vec{q}$ rotates between the three orthogonal vectors ${\vec{n}}_i$, corresponding to the directions of maximal $\left({\vec{n}}_1\right)$ and minimal $\left({\vec{n}}_3\right)$ spread of the H ground-state wave function in a dislocation core. The peak along each ${\vec{n}}_i$ indicates that the excited state is $p$-like along that direction. The form factors for bulk excited states which maximize overlap with the dislocation core states shows a similar peak shape with direction. The height of the peaks corresponds to the spreading of the excited states along those directions in comparison with the ground state: increased spreading of the excited states along the particular direction decreases the peak height.