Structural properties of ${}^4\mathrm{He}_N$ ($N=2–10$) clusters for different potential models at the physical point and at unitarity

Since the ${}^4\mathrm{He}$ dimer supports only one weakly bound state with an average interatomic distance much larger than the van der Waals length and no deeply bound states, ${}^4\mathrm{He}_N$ clusters with $N>2$ are a paradigmatic model system with which to explore foundational concepts such as large $s$-wave scattering length universality, van der Waals universality, Efimov physics, and effective field theories. This work presents structural properties such as the pair and triple distribution functions, the hyper-radial density, the probability to find the $N\mathrm{th}$ particle at a given distance from the center of mass of the other $N-1$ atoms, and selected contacts. The kinetic energy release, which can be measured via Coulomb explosion in dedicated size-selected molecular beam experiments—at least for small $N$ – is also presented. The structural properties are determined for three different realistic ${}^4\mathrm{He}-{}^4\mathrm{He}$ interaction potentials and contrasted with those for an effective low-energy potential model from the literature that reproduces the energies of ${}^4\mathrm{He}_N$ clusters in the ground state for $N=2$ to $N=\infty$ at the $\gtrsim$95% level with just four input parameters. The study is extended to unitarity (infinite $s$-wave scattering length) by artificially weakening the interaction potentials. In addition to contributing to the characterization of small bosonic helium quantum droplets, our study provides insights into the effective low-energy theory's predictability of various structural properties.

Figure 1

${\rho }^2{P}_N^{\left(\text{jacobi}\right)}\left(\rho \right)$ for $N=3–10$ at the physical point (top row) and at unitarity (bottom row). The solid lines in (a)–(d) show ${\rho }^2{P}_N^{\left(\text{jacobi}\right)}\left(\rho \right)$ for model IB. At $\rho =20 a_0$, the curves are ordered, from top to bottom, from the smallest $N$ (the curve for $N=3$ is green) to the largest $N$ (the curve for $N=10$ is dark red). Note that the data shown in the first and second columns are identical, but that the $x$ and $y$ scales differ. The dotted lines in (b) and (d) show the asymptotic behavior $A_{N}exp(-2{\kappa }_N\rho )$, using the numerically determined binding momentum ${\kappa }_N$ and treating the “normalization constant” $A_N$ as a fitting parameter. The solid and dotted lines in (e) and (f) (third column) show ${\rho }^2{P}_N^{\left(\text{jacobi}\right)}\left(\rho \right)$ for model IA and model II, respectively. The agreement between the dotted and solid lines is excellent at large $\rho$ and deteriorates for smaller $\rho$. The deterioration is due to the inability of the low-energy model to fully capture the small length scale correlations. The color scheme used here is also used in Figs. 2, 3, 4, 6, and 7, and in the Supplemental Material [82], Figs. S3 and S4. The layout used here, i.e., the top row showing results at the physical point and the bottom row showing results at unitarity, is also used in Figs. 2, 5, 6, 7, and in the Supplemental Material [82], Fig. S4.

Figure 2

$r^2{P}_N^{\left(2\right)}\left(r\right)$ for $N=2–10$ at the physical point (top row) and for $N=3–10$ at unitarity (bottom row). The solid lines in (a)–(d) show $r^2{P}_N^{\left(2\right)}\left(r\right)$ for model IB. Note that the data shown in the first and second columns are identical: the first column shows $r$ in units of $a_0$ and the second column shows $r$ in units of $r_{\text{vdW}}$, focusing on the small-$r$ region. It can be seen that the scaled pair distribution functions for different $N$ collapse approximately for $r\approx r_{\text{vdW}}$. The solid and dotted lines in (e) and (f) (third column) show $r^2{P}_N^{\left(2\right)}\left(r\right)$ for model IA and model II, respectively. Differences are most pronounced in the $r\lesssim 20 a_0$ region. The color scheme is the same as in Fig. 1.

Figure 3

$r^2{P}_N^{\left(2\right)}\left(r\right)/C_N^{\left(2\right)}, N=2–10$, for two realistic interaction potentials at the physical point. The sets of dash-dotted and solid lines show results for model IA and model IB, respectively. The color scheme is the same as in Fig. 1. The thin black dash-dotted and solid lines show the universal van der Waals function $|{\phi }_{\text{vdW}}{\left(r\right)|}^2$, given by Eq. (26), for model IA and model IB, respectively (these two models are characterized by slightly different $r_{\text{vdW}}$); the normalization constant $B$ is adjusted by fitting $|{\phi }_{\text{vdW}}{\left(r\right)|}^2$ to $r^2{P}_2^{\left(2\right)}\left(r\right)$, including $r$ values ($r\lesssim 2r_{\text{vdW}}$) for which $P_2^{\left(2\right)}\left(r\right)$ takes values that are larger than 5% and smaller than the maximum of $P_2^{\left(2\right)}\left(r\right)$ for model IA and smaller than 95% of the maximum of $P_2^{\left(2\right)}\left(r\right)$ for model IB, respectively.

Figure 4

Triple correlations and $(2+1)$ contact. (a) The solid and dash-dotted lines show ${\left({\rho }_3\right)}^2{P}_N^{(3,\text{jacobi})}\left({\rho }_3\right)/C_N^{(2+1)}$ for the realistic HFD-HE2 potential (model IA) and the effective low-energy potential (model II), respectively, at the physical point for $N=3–10$. The color scheme is the same as in Fig. 1. (b) The squares, circles, and triangles show the pair contact $C_N^{(2+1)}$ at the physical point as a function of $N$ for models IA, IB, and II, respectively. (c) The squares, circles, and triangles show the pair contact $C_N^{(2+1)}$ at unitarity as a function of $N$ for models IA, IB, and II, respectively.

Figure 5

$P_N^{(3,\text{shape})}(\overline{x},\overline{y})$ for model IB at the physical point (top row) and at unitarity (bottom row). The first, second, and third columns show results for $N=3, N=4$, and $N=10$, respectively. The color bar on the right applies to all six panels. Since the triangles are oriented and normalized such that one particle sits at $(\overline{x},\overline{y})=(–1/2,0)$ and the other at $(\overline{x},\overline{y})=(+1/2,0)$ (with the interparticle distance vector corresponding to the largest distance being oriented along the $\pm \overline{x}$ axis), the regions in the top left and top right of the panels are excluded by construction.

Figure 6

Hyper-radial properties for $N=3–10$ at the physical point (top row) and at unitarity (bottom row). The first column shows $P_N^{\left(\text{hyper}\right)}\left({\rho }_N\right)$ for model IA. The second column shows the (approximate) effective potential curves $V_{\text{eff}}\left({\rho }_N\right)$ for model IA, calculated using Eq. (27). For comparison, the third column shows $V_{\text{eff}}\left({\rho }_N\right)$ for model II (the model II plots are made using the van der Waals length for model IA as a scale). The color scheme is the same as in Fig. 1.

Figure 7

KER for $N=3–10$ at the physical point (top row) and at unitarity (bottom row). The first, second, and third columns show the KER for models IA, IB, and II, respectively. The color scheme is the same as in Fig. 1.