Figure 1

Interatomic potentials for He-He and He-Ca systems.Reuse & Permissions

Figure 2

The lowest hyperspherical potential curves for each of the triatomic systems, ${}^3$He-${}^3$He-${}^{40}$Ca, ${}^3$He-${}^4$He-${}^{40}$Ca, and ${}^4$He-${}^4$He-${}^{40}$Ca trimers. The curves of both the ${}^3$He-${}^4$He-${}^{40}$Ca and ${}^4$He-${}^4$He-${}^{40}$Ca trimers converge to −5.63 K, the binding energy of the bound state of the ${}^4$He-${}^{40}$Ca dimer. The curve of ${}^3$He-${}^3$He-${}^{40}$Ca trimer converges to −4.74K, the binding energy of the bound state of ${}^3$He-${}^{40}$Ca dimer. Both $U_0$($R$) and $U_0$($R$) $+$ $W_{00}$($R$) are presented in the inset for the ${}^4$He-${}^4$He-${}^{40}$Ca trimer.Reuse & Permissions

Figure 3

The wave functions for the bound states of the ${}^3$He-${}^3$He-${}^{40}$Ca, ${}^3$He-${}^4$He-${}^{40}$Ca, and ${}^4$He-${}^4$He-${}^{40}$Ca trimers.Reuse & Permissions

Figure 4

Two-dimensional contour plots of the potential surface $V(R,\phi ,\theta )$ and the probability densities of the channel functions at fixed hyper-radius $R$ as functions of the hyperangles $\phi$ and $\theta$. The first row are the potential surfaces at $R$ corresponding to the minimum of the hyperspherical potential curves for ${}^3$He-${}^3$He-${}^{40}$Ca, ${}^4$He-${}^4$He-${}^{40}$Ca, and ${}^3$He-${}^4$He-${}^{40}$Ca, respectively, and the second row is the probability densities at $R$ corresponding to the potential surfaces in the first row. The third and fifth rows are the three-body potentials of the three trimers at $R$ $=$ 100 a.u. and $R$ $=$ 400 a.u., respectively, and the fourth and sixth rows are the probability densities at $R$ = 100 a.u. and $R$ $=$ 400 a.u. corresponding to the potential surfaces in the third and fifth rows, respectively. The red points on the potential surfaces are the lowest points of the potential surfaces and the red points on the probability densities are the highest points of the probability densities. The black points on the figure are the coalescence points for ${}^3$He-${}^{40}$Ca or ${}^4$He-${}^{40}$Ca. The two ${}^3$He-${}^{40}$Ca coalescence points for the symmetric system of ${}^3$He-${}^3$He-${}^{40}$Ca trimer are located at $(\phi ,\theta )=(0.239\pi ,0)$ and $(\phi ,\theta )=(0.239\pi ,\pi )$. The two ${}^4$He-${}^{40}$Ca coalescence points for the ${}^4$He-${}^4$He-${}^{40}$Ca trimer are located at $(\phi ,\theta )=(0.235\pi ,0)$ and $(\phi ,\theta )=(0.235\pi ,\pi )$. For the asymmetric system of ${}^3$He-${}^4$He-${}^{40}$Ca, the ${}^3$He-${}^{40}$Ca coalescence point is located at $(\phi ,\theta )=(0.260\pi ,0)$ and the ${}^4$He-${}^{40}$Ca coalescence point is at $(\phi ,\theta )=(0.215\pi ,\pi )$. The coalescence locations of two helium atoms are line $\phi =0.5\pi$ on the figure.Reuse & Permissions

Figure 5

The most probable geometries of ${}^3$He-${}^3$He-${}^{40}$Ca, ${}^4$He-${}^4$He-${}^{40}$Ca, and ${}^3$He-${}^4$He-${}^{40}$Ca trimers at $R$ corresponding to the minimum of the hyperspherical potential curves, $R$ $=$ 100 a.u., and $R$ $=$ 400 a.u.Reuse & Permissions

Figure 6

The averaged interatomic separation $〈{\rho }_1〉$ between two ${}^4$He atoms and the averaged distance $〈{\rho }_2〉$ of the Ca atom measured from the center of mass of the ${}^4$He-${}^4$He pair for the ${}^4$He-${}^4$He-${}^{40}$Ca trimer.Reuse & Permissions