${}^4\mathrm{He}$ adsorption on a ${\text{H}}_2$-plated ${\text{C}}_{20}$ molecular surface: The formation of helium buckyballs

We perform path-integral Monte Carlo calculations to study the adsorption of ${}^4\mathrm{He}$ atoms on a ${\text{H}}_2$-plated ${\text{C}}_{20}$ molecular surface. It is found that 32 ${\text{H}}_2$ molecules form a complete solid layer on ${\text{C}}_{20}$, where each ${\text{H}}_2$ molecule is located either above one of the 12 pentagon centers or above one of the 20 carbon atoms. The angular density profiles of the first ${}^4\mathrm{He}$ layer on the (${\text{H}}_2$)${}_{32}$-${\text{C}}_{20}$ surface reveal different quantum states as the number of ${}^4\mathrm{He}$ atoms $N$ varies. Especially, the helium layer exhibits an icosidodecahedron structure for $N=30$, where each ${}^4\mathrm{He}$ atom is located at one of the vertices of 20 corner-sharing triangles. While the ${}^4\mathrm{He}$ density peaks for $N=60$ constitute a truncated icosahedron with 12 pentagonal and 20 hexagonal faces, the additional atoms beyond $N=60$ are found to be placed at the hexagon centers of the truncated icosahedron to form a hexakis truncated icosahedron for $N=80$. The superfluid response of the ${}^4\mathrm{He}$ layer at a temperature of $T=0.6$ K is found to be completely quenched for $N=30$ and to be significantly suppressed for $N=60$ and 80, reflecting the formation of compact buckyball structures.

Figure 1

(a) Angular and (b) radial density distributions of 32 ${\text{H}}_2$ molecules adsorbed on ${\text{C}}_{20}$. The vertical axis and the horizontal axis in panel (a) correspond to the cosine of the polar angle $\theta$ and the azimuthal angle $\varphi$ while the characters $C$ and $P$ represent the angular positions of the carbon atoms and those of the pentagon centers, respectively. The solid line in panel (b) represents the total density distribution and the dotted (dot-dashed) line corresponds to the density distribution of 12 (20) ${\text{H}}_2$ molecules located at the $P$ sites ($C$ sites).

Figure 2

(a) Radial density distributions of 32 ${\text{H}}_2$ molecules (dotted line) and $N$ ${}^4\mathrm{He}$ atoms (solid and dashed lines) in the (${}^4\mathrm{He}$)${}_N$-(${\text{H}}_2$)${}_{32}$-${\text{C}}_{20}$ complex as a function of the distance from the center of ${\text{C}}_{20}$ and (b) the total energy per ${}^4\mathrm{He}$ atom of the ${}^4\mathrm{He}$ layer adsorbed on the ${\text{H}}_2$-plated ${\text{C}}_{20}$ surface in units of kelvin. The PIMC calculations were done at $T=0.6$ K and the dotted line in panel (b) is just a guide to the eye.

Figure 3

Contour plots of angular density distributions of $N$ ${}^4\mathrm{He}$ atoms adsorbed on the surface of an (${\text{H}}_2$)${}_{32}$-${\text{C}}_{20}$ complex for (a) $N=10$, (b) $N=30$, (c) $N=60$, and (d) $N=80$. The horizontal axis corresponds to the azimuthal angle $\varphi$ and the vertical one to the cosine of the polar angle $\theta$. The white dots represent the peak positions of the underlying ${\text{H}}_2$ layer and the characters $C$ and $P$ correspond to the angular positions of the carbon atoms and those of the pentagon centers of ${\text{C}}_{20}$, respectively. The PIMC calculations were performed at $T=0.6$ K and all contour plots are in the same color scale denoted by the color table in the upper right-hand corner.

Figure 4

Three-dimensional plots of buckyball structures of $N$ ${}^4\mathrm{He}$ atoms [red (gray) solid dots] in the (${}^4\mathrm{He}$)${}_N$-(${\text{H}}_2$)${}_{32}$-${\text{C}}_{20}$ complexes for (a) $N=30$ (icosidodecahedron), (b) $N=60$ (truncated icosahedron), and (c) $N=80$ (hexakis truncated icosahedron). The blue open dots and the gray-colored inner structure represent 32 ${\text{H}}_2$ molecules and a ${\text{C}}_{20}$ molecule, respectively.

Figure 5

Superfluid fractions of the ${}^4\mathrm{He}$ layer on the surface of the (${\text{H}}_2$)${}_{32}$-${\text{C}}_{20}$ complex as a function of the number of ${}^4\mathrm{He}$ atoms $N$ at $T=0.6$ and $1.2$ K. The dotted lines are just a guide to the eye.